Take the primary term in the denominator and the quadratic term without real root as factorization, and then use the undetermined coefficient method to find the corresponding numerator.

Personally feel that the subject may not understand what to do after decomposition. For example, the first problem no longer needs to be decomposed.

There is no problem in visually checking the integral of the term once. The following is the integral method of quadratic term without real root.

1) If the numerator is also a quadratic term (even higher), use polynomial long division. Solve a polynomial from the molecule and integrate it separately.

2) If the molecule is a linear term, decompose the molecule into the form of derivative+constant of the parent.

The first part is processed by rounding and differentiation, and the second part is processed by arc tangent.

The first question is for reference.